A note on the approximation properties of frames of general form (Q1810190)

Let \(H\) be a separable real Hilbert space. A system \(\phi=\{\phi_j;\;j\in \mathbb{N}\}\) is called a frame of \(H\), if for all \(f\in H\) \[ a\| f\|^2\leq \sum_{j=1} (f,\varphi_j)^2\leq b\| f\|^2 \] with \(0<a\leq b<\infty\). Let \(\{\widetilde\varphi_j;\;j\in\mathbb{N}\}\) be the dual frame of \(\phi\). Further let \[ W_N= \left\{f=\sum^N_{k=1}\varepsilon_k\psi_k;\;\varepsilon_k=\pm 1\right\} \] be the set of vertices of an \(N\)-dimensional cube, where \(\{\psi_k;\;=1,\dots,N\}\) an orthonormal system. The authors apply the geometric approach to best \(m\)-term approximations proposed by the first author [Tr. Mat. Inst. Steklova 172, 187--191 (1985; Zbl 0581.42018)] to find a lower bound of the quantities \[ \sigma_m(W_N,\phi)=\sup_{f\in W_N} \left(\inf\left\| f-\sum_{j\in\Lambda}(f,\widetilde \varphi_j)\varphi_j\right\|\right), \] where \(\Lambda\) is an arbitrary subset of \(\mathbb{N}\) with at most \(m\) elements.